The truth of the matter
Trying to remove hiss, hum, rumble, and similar noise, can be likened to trying to remove the milk from a cup of tea or coffee. Often, the noise can be reduced without degrading the underlying audio to an unacceptable extent. Sometimes the result will sound exactly as if the noise were actually removed, sometimes an objectionable torrent of noise will be reduced to acceptable background noise. Listen to the samples (Capercaille, Ted Weems, Percy Grainger) to see what can be achieved in reducing hum and hiss using the software on this site.
• Frequency domain based – used by the DeNoise application for hiss and other general broadband noise problems.
• Non-linear filter and wavelet based – used by the DeNoiseLF application for the treatment of hum, rumble, and other low-frequency noise.
Although the removed "noise" will inevitably be correlated somewhat with the desirable audio, unpleasant processing artifacts should be minimal with reasonably chosen settings. But do remember that there is no way to perfectly un-mix what has been mixed!